Convert Cartesian to Polar Coordinates

Point in the Polar and Cartesian System

This example shows how to convert a point from the Cartesian Coordinate System to the Polar Coordinate System. The general conversion is shown in the two equations below.

Convert Cartesian to Polar Example

For example, to convert the cartesian point to polar coordinates. First we solve for the hypotenuse length of the right triangle to find the length of the radius:

Then use the arctangent function to find the angle:

The cartesian point is equivalent to the polar point . Note, if the same calculation is performed with a calculator set to degrees instead of radians the point would be . The right triangle formed by both points is shown below:

Point in the Polar and Cartesian System

Explanation

Both systems describe the position of a point in space. A point in the Polar Coordinate System is defined in terms of a radius and an angle: . A point in the Cartesian Coordinate System is defined in terms of a and component: . Both define the point relative to the origin of the system. Geometrically, the two points can be described by the right triangle below.

Polar Coordinates Point

The pythagorean theorem relates the squared sides together on a right-triangle. Since the component corresponds to the adjacent side of the right triangle and the component corresponds to the opposite side, the equation can be rearranged to give the length of the hypotenuse which corresponds to the length of the radius in a polar coordinate.

The arctangent function returns the angle of a right-triangle given the ratio of its opposite side over its adjacent side.